# Phase space plot

• A
Aatifa

## Main Question or Discussion Point

Hi, i would like to know how can i plot a three dimentionnal phase space (mathematica), for this kind of differential equations:
x'= (z^2(x-4y+z)+y^2(x+z)+z(x*y-x^2))/(z^2+x^2-2)
y'=y(y-3)+z(4y-z)+3(1-x^2)-2(y-3z)(z^2(x-4y+z)+y^2(x+z)+z(x*y-x^2))/x(z^2+x^2-2)
z'=2y-z(z^2(x-4y+z)+y^2(x+z)+z(x*y-x^2))/x(z^2+x^2-2)
do i need to solve numerically this system and then plot its solution?

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S.G. Janssens
The phase space itself would be $\mathbb{R}^3$ (or: a subset of $\mathbb{R}^3$), so for that there is not much to plot. (Just three axes.)

You could plot the vector field defined by the right-hand side (I think Mathematica can do that?) or you could plot some "typical" solutions. For the former, you do not need to solve the ODE, for the latter you do need to solve it indeed. A combination is also possible: Plot the vector field and superimpose on this a plot of some typical solutions starting from various initial conditions.

Aatifa
Hi, thank you for your reply. Actually what i am trying to do is to plot the phase space of my dynamical system and superimpose on this plot the critical points of this system in order to study the stability of these critical points. Here is the result of the plot, unfortunatelly i didn't get any useful information from it.
I would like to know what did you mean by "typical" solutions?

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S.G. Janssens
Actually what i am trying to do is to plot the phase space of my dynamical system and superimpose on this plot the critical points of this system in order to study the stability of these critical points. Here is the result of the plot, unfortunatelly i didn't get any useful information from it.
Probably you already calculated the eigenvalues of the linearization at the critical points? Could you draw any conclusions regarding (local) stability from that?
I would like to know what did you mean by "typical" solutions?
I'm sorry, "typical" is indeed an ill-defined notion here. I meant to plot a couple of orbits through various initial conditions in the phase space to get an idea of what they look like. Likely this would work in Mathematica with some experimenting, but I am not familiar with that software. Alternatively, there are also more specialized (free) toolboxes that can help you with this and subsequent analysis. One of those is MatCont (software, Scholarpedia page), which is developed academically. (It requires MATLAB, though it might also run in Octave.)

Aatifa
Probably you already calculated the eigenvalues of the linearization at the critical points? Could you draw any conclusions regarding (local) stability from that?

Yes, i did, actually i found some sadlle and stable points. But i was wondering if i could confirm this result by ploting the phase space.

I'm sorry, "typical" is indeed an ill-defined notion here. I meant to plot a couple of orbits through various initial conditions in the phase space to get an idea of what they look like. Likely this would work in Mathematica with some experimenting, but I am not familiar with that software. Alternatively, there are also more specialized (free) toolboxes that can help you with this and subsequent analysis. One of those is MatCont (software, Scholarpedia page), which is developed academically. (It requires MATLAB, though it might also run in Octave.)
thank you for your clarifications about my question, i will try to use one of these softwares that you have mensionned above.